What Is The Term Used To Describe The Distribution Of A Data Set With One Mode

Mathematical part for the probability a given outcome occurs in an experiment

In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment.^{[ane] [2]} It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space).^[3]

For case, if 10 is used to denote the event of a coin toss ("the experiment"), then the probability distribution of Ten would take the value 0.5 (1 in 2 or i/2) for X = heads, and 0.five for X = tails (assuming that the coin is fair). Examples of random phenomena include the atmospheric condition weather condition at some future date, the summit of a randomly selected person, the fraction of male students in a school, the results of a survey to be conducted, etc.^[four]

Introduction [edit]

The probability mass function (pmf) ${\displaystyle p(S)}$ specifies the probability distribution for the sum ${\displaystyle S}$ of counts from two die. For example, the effigy shows that ${\displaystyle p(11)=2/36=1/18}$ . The pmf allows the computation of probabilities of events such as ${\displaystyle P(X>9)=1/12+1/18+1/36=1/half-dozen}$ , and all other probabilities in the distribution.

A probability distribution is a mathematical description of the probabilities of events, subsets of the sample space. The sample space, often denoted by ${\displaystyle \Omega }$ , is the set of all possible outcomes of a random miracle being observed; it may be whatever fix: a set up of existent numbers, a prepare of vectors, a prepare of arbitrary non-numerical values, etc. For example, the sample infinite of a coin flip would be Ω = {heads, tails}.

To ascertain probability distributions for the specific case of random variables (so the sample space tin be seen as a numeric fix), it is common to distinguish between detached and absolutely continuous random variables. In the discrete example, it is sufficient to specify a probability mass office ${\displaystyle p}$ assigning a probability to each possible outcome: for example, when throwing a fair die, each of the 6 values 1 to 6 has the probability 1/half dozen. The probability of an consequence is then defined to be the sum of the probabilities of the outcomes that satisfy the event; for example, the probability of the consequence "the dice rolls an even value" is

$${\displaystyle p(two)+p(4)+p(six)=1/six+1/6+i/half-dozen=1/2.}$$

In contrast, when a random variable takes values from a continuum then typically, whatsoever private outcome has probability nix and only events that include infinitely many outcomes, such as intervals, can have positive probability. For example, consider measuring the weight of a slice of ham in the supermarket, and presume the calibration has many digits of precision. The probability that it weighs exactly 500 g is cypher, as information technology will near likely have some not-nothing decimal digits. Nevertheless, one might demand, in quality control, that a package of "500 thou" of ham must weigh between 490 g and 510 g with at least 98% probability, and this need is less sensitive to the accurateness of measurement instruments.

The left shows the probability density office. The right shows the cumulative distribution office, for which the value at a equals the area under the probability density curve to the left of a.

Absolutely continuous probability distributions can be described in several ways. The probability density function describes the infinitesimal probability of whatsoever given value, and the probability that the issue lies in a given interval can be computed by integrating the probability density office over that interval.^[5] An alternative clarification of the distribution is past means of the cumulative distribution function, which describes the probability that the random variable is no larger than a given value (i.e., ${\displaystyle P(10<x)}$ for some ${\displaystyle x}$ ). The cumulative distribution function is the expanse under the probability density function from ${\displaystyle -\infty }$ to ${\displaystyle x}$ , equally described by the film to the right.^[six]

General definition [edit]

A probability distribution can be described in various forms, such as by a probability mass function or a cumulative distribution part. 1 of the most general descriptions, which applies for admittedly continuous and discrete variables, is by means of a probability function ${\displaystyle P\colon {\mathcal {A}}\to \mathbb {R} }$ whose input infinite ${\displaystyle {\mathcal {A}}}$ is related to the sample space, and gives a real number probability every bit its output.^[7]

The probability function ${\displaystyle P}$ can take as argument subsets of the sample space itself, as in the coin toss example, where the function ${\displaystyle P}$ was divers so that P(heads) = 0.five and P(tails) = 0.5. Notwithstanding, considering of the widespread use of random variables, which transform the sample space into a set of numbers (e.thousand., ${\displaystyle \mathbb {R} }$ , ${\displaystyle \mathbb {N} }$ ), it is more common to study probability distributions whose argument are subsets of these detail kinds of sets (number sets),^[8] and all probability distributions discussed in this article are of this type. It is common to denote every bit ${\displaystyle P(X\in E)}$ the probability that a certain variable ${\displaystyle Ten}$ belongs to a certain consequence ${\displaystyle E}$ .^{[four] [9]}

The above probability function only characterizes a probability distribution if it satisfies all the Kolmogorov axioms, that is:

1. ${\displaystyle P(Ten\in E)\geq 0\;\forall Eastward\in {\mathcal {A}}}$ , and so the probability is not-negative
2. ${\displaystyle P(Ten\in East)\leq 1\;\forall East\in {\mathcal {A}}}$ , so no probability exceeds ${\displaystyle 1}$
3. ${\displaystyle P(X\in \bigsqcup _{i}E_{i})=\sum _{i}P(10\in E_{i})}$ for any disjoint family of sets ${\displaystyle \{E_{i}\}}$

The concept of probability function is made more rigorous by defining information technology as the element of a probability space ${\displaystyle (X,{\mathcal {A}},P)}$ , where ${\displaystyle X}$ is the set of possible outcomes, ${\displaystyle {\mathcal {A}}}$ is the set of all subsets ${\displaystyle Due east\subset X}$ whose probability can be measured, and ${\displaystyle P}$ is the probability part, or probability measure, that assigns a probability to each of these measurable subsets ${\displaystyle E\in {\mathcal {A}}}$ .^[10]

Probability distributions normally belong to one of two classes. A detached probability distribution is applicable to the scenarios where the fix of possible outcomes is discrete (eastward.g. a money toss, a roll of a die) and the probabilities are encoded by a discrete listing of the probabilities of the outcomes; in this instance the discrete probability distribution is known as probability mass role. On the other hand, absolutely continuous probability distributions are applicative to scenarios where the prepare of possible outcomes tin can take on values in a continuous range (e.k. real numbers), such as the temperature on a given day. In the absolutely continuous case, probabilities are described by a probability density role, and the probability distribution is past definition the integral of the probability density office.^{[4] [5] [nine]} The normal distribution is a ordinarily encountered admittedly continuous probability distribution. More complex experiments, such as those involving stochastic processes divers in continuous time, may demand the use of more general probability measures.

A probability distribution whose sample space is i-dimensional (for example real numbers, list of labels, ordered labels or binary) is called univariate, while a distribution whose sample infinite is a vector space of dimension ii or more is chosen multivariate. A univariate distribution gives the probabilities of a unmarried random variable taking on various different values; a multivariate distribution (a joint probability distribution) gives the probabilities of a random vector – a list of two or more random variables – taking on various combinations of values. Important and usually encountered univariate probability distributions include the binomial distribution, the hypergeometric distribution, and the normal distribution. A unremarkably encountered multivariate distribution is the multivariate normal distribution.

Besides the probability role, the cumulative distribution function, the probability mass role and the probability density function, the moment generating function and the characteristic function also serve to identify a probability distribution, as they uniquely decide an underlying cumulative distribution function.^[11]

The probability density function (pdf) of the normal distribution, likewise called Gaussian or "bell curve", the well-nigh important admittedly continuous random distribution. As notated on the effigy, the probabilities of intervals of values correspond to the expanse under the curve.

Terminology [edit]

Some key concepts and terms, widely used in the literature on the topic of probability distributions, are listed below.^[i]

Basic terms [edit]

Discrete probability distributions [edit]

• Discrete probability distribution: for many random variables with finitely or countably infinitely many values.
• Probability mass role (pmf): function that gives the probability that a discrete random variable is equal to some value.
• Frequency distribution: a table that displays the frequency of various outcomes in a sample.
• Relative frequency distribution: a frequency distribution where each value has been divided (normalized) past a number of outcomes in a sample (i.e. sample size).
• Categorical distribution: for detached random variables with a finite prepare of values.

Absolutely continuous probability distributions [edit]

• Absolutely continuous probability distribution: for many random variables with uncountably many values.
• Probability density office (pdf) or Probability density: part whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) tin be interpreted every bit providing a relative likelihood that the value of the random variable would equal that sample.

[edit]

Cumulative distribution function [edit]

In the special case of a real-valued random variable, the probability distribution can equivalently exist represented past a cumulative distribution part instead of a probability measure. The cumulative distribution function of a random variable ${\displaystyle X}$ with regard to a probability distribution ${\displaystyle p}$ is divers as

$${\displaystyle F(ten)=P(X\leq 10).}$$

The cumulative distribution part of any real-valued random variable has the properties:

Conversely, whatever function ${\displaystyle F:\mathbb {R} \to \mathbb {R} }$ that satisfies the outset four of the backdrop in a higher place is the cumulative distribution function of some probability distribution on the real numbers.^[13]

Whatsoever probability distribution can exist decomposed as the sum of a discrete, an absolutely continuous and a atypical continuous distribution,^[14] and thus any cumulative distribution office admits a decomposition equally the sum of the three according cumulative distribution functions.

Discrete probability distribution [edit]

The probability mass function of a detached probability distribution. The probabilities of the singletons {1}, {3}, and {vii} are respectively 0.2, 0.5, 0.3. A set not containing any of these points has probability zero.

The cdf of a discrete probability distribution, ...

... of a continuous probability distribution, ...

... of a distribution which has both a continuous part and a discrete part.

A discrete probability distribution is the probability distribution of a random variable that can take on only a countable number of values^[15] (almost surely)^[xvi] which means that the probability of any event ${\displaystyle E}$ can be expressed every bit a (finite or countably infinite) sum:

$${\displaystyle P(X\in E)=\sum _{\omega \in A}P(X=\omega ),}$$

where ${\displaystyle A}$ is a countable set. Thus the detached random variables are exactly those with a probability mass office ${\displaystyle p(x)=P(X=x)}$ . In the case where the range of values is countably infinite, these values accept to decline to cipher fast enough for the probabilities to add upward to 1. For example, if ${\displaystyle p(north)={\tfrac {one}{two^{n}}}}$ for ${\displaystyle north=one,two,...}$ , the sum of probabilities would be ${\displaystyle ane/2+1/4+1/8+\dots =1}$ .

A discrete random variable is a random variable whose probability distribution is discrete.

Well-known detached probability distributions used in statistical modeling include the Poisson distribution, the Bernoulli distribution, the binomial distribution, the geometric distribution, the negative binomial distribution and categorical distribution.^[3] When a sample (a ready of observations) is drawn from a larger population, the sample points have an empirical distribution that is detached, and which provides information about the population distribution. Additionally, the discrete uniform distribution is commonly used in figurer programs that make equal-probability random selections between a number of choices.

Cumulative distribution function [edit]

A existent-valued detached random variable can equivalently be defined as a random variable whose cumulative distribution office increases only past jump discontinuities—that is, its cdf increases merely where it "jumps" to a higher value, and is abiding in intervals without jumps. The points where jumps occur are precisely the values which the random variable may accept. Thus the cumulative distribution function has the course

$${\displaystyle F(x)=P(X\leq x)=\sum _{\omega \leq x}p(\omega ).}$$

Note that the points where the cdf jumps always form a countable set; this may exist any countable set and thus may even be dumbo in the real numbers.

Dirac delta representation [edit]

A detached probability distribution is often represented with Dirac measures, the probability distributions of deterministic random variables. For whatever outcome ${\displaystyle \omega }$ , let ${\displaystyle \delta _{\omega }}$ be the Dirac measure concentrated at ${\displaystyle \omega }$ . Given a discrete probability distribution, there is a countable set ${\displaystyle A}$ with ${\displaystyle P(Ten\in A)=ane}$ and a probability mass function ${\displaystyle p}$ . If ${\displaystyle Eastward}$ is any event, then

$${\displaystyle P(X\in E)=\sum _{\omega \in A}p(\omega )\delta _{\omega }(Eastward),}$$

or in short,

$${\displaystyle P_{10}=\sum _{\omega \in A}p(\omega )\delta _{\omega }.}$$

Similarly, discrete distributions can be represented with the Dirac delta role equally a generalized probability density role ${\displaystyle f}$ , where

$${\displaystyle f(ten)=\sum _{\omega \in A}p(\omega )\delta (x-\omega ),}$$

which means

$${\displaystyle P(X\in E)=\int _{East}f(x)\,dx=\sum _{\omega \in A}p(\omega )\int _{E}\delta (ten-\omega )=\sum _{\omega \in A\cap Due east}p(\omega )}$$

for whatsoever event ${\displaystyle E.}$ ^[17]

Indicator-function representation [edit]

For a discrete random variable ${\displaystyle X}$ , let ${\displaystyle u_{0},u_{ane},\dots }$ be the values information technology can take with non-zero probability. Denote

$${\displaystyle \Omega _{i}=X^{-one}(u_{i})=\{\omega :X(\omega )=u_{i}\},\,i=0,ane,2,\dots }$$

These are disjoint sets, and for such sets

$${\displaystyle P\left(\bigcup _{i}\Omega _{i}\correct)=\sum _{i}P(\Omega _{i})=\sum _{i}P(X=u_{i})=1.}$$

It follows that the probability that ${\displaystyle 10}$ takes whatsoever value except for ${\displaystyle u_{0},u_{ane},\dots }$ is zero, and thus one can write ${\displaystyle Ten}$ every bit

$${\displaystyle X(\omega )=\sum _{i}u_{i}1_{\Omega _{i}}(\omega )}$$

except on a set of probability zero, where ${\displaystyle 1_{A}}$ is the indicator function of ${\displaystyle A}$ . This may serve as an alternative definition of detached random variables.

One-bespeak distribution [edit]

A special example is the discrete distribution of a random variable that can have on only one stock-still value; in other words, information technology is a deterministic distribution. Expressed formally, the random variable ${\displaystyle X}$ has a one-bespeak distribution if it has a possible effect ${\displaystyle x}$ such that ${\displaystyle P(X{=}x)=1.}$ ^[xviii] All other possible outcomes then have probability 0. Its cumulative distribution office jumps immediately from 0 to 1.

Admittedly continuous probability distribution [edit]

An admittedly continuous probability distribution is a probability distribution on the real numbers with uncountably many possible values, such as a whole interval in the real line, and where the probability of any event tin can be expressed as an integral.^[19] More precisely, a existent random variable ${\displaystyle X}$ has an admittedly continuous probability distribution if in that location is a function ${\displaystyle f:\mathbb {R} \to [0,\infty ]}$ such that for each interval ${\displaystyle [a,b]\subset \mathbb {R} }$ the probability of ${\displaystyle X}$ belonging to ${\displaystyle [a,b]}$ is given by the integral of ${\displaystyle f}$ over ${\displaystyle I}$ :^{[20] [21]}

$${\displaystyle P\left(a\leq X\leq b\right)=\int _{a}^{b}f(ten)\,dx.}$$

This is the definition of a probability density office, and then that absolutely continuous probability distributions are exactly those with a probability density office. In particular, the probability for ${\displaystyle X}$ to take whatever single value ${\displaystyle a}$ (that is, ${\displaystyle a\leq X\leq a}$ ) is zero, because an integral with coinciding upper and lower limits is always equal to naught. If the interval ${\displaystyle [a,b]}$ is replaced by any measurable set ${\displaystyle A}$ , the according equality yet holds:

$${\displaystyle P(X\in A)=\int _{A}f(x)\,dx.}$$

An admittedly continuous random variable is a random variable whose probability distribution is admittedly continuous.

There are many examples of absolutely continuous probability distributions: normal, compatible, chi-squared, and others.

Cumulative distribution function [edit]

Absolutely continuous probability distributions as defined above are precisely those with an absolutely continuous cumulative distribution role. In this case, the cumulative distribution function ${\displaystyle F}$ has the class

$${\displaystyle F(ten)=P(X\leq x)=\int _{-\infty }^{x}f(t)\,dt}$$

where ${\displaystyle f}$ is a density of the random variable ${\displaystyle 10}$ with regard to the distribution ${\displaystyle P}$ .

Note on terminology: Absolutely continuous distributions ought to be distinguished from continuous distributions, which are those having a continuous cumulative distribution function. Every absolutely continuous distribution is a continuous distribution simply the antipodal is not true, there be singular distributions, which are neither absolutely continuous nor discrete nor a mixture of those, and do not have a density. An example is given past the Cantor distribution. Some authors however use the term "continuous distribution" to denote all distributions whose cumulative distribution office is admittedly continuous, i.e. refer to absolutely continuous distributions equally continuous distributions.^[4]

For a more general definition of density functions and the equivalent absolutely continuous measures see absolutely continuous measure out.

Kolmogorov definition [edit]

In the measure-theoretic formalization of probability theory, a random variable is defined as a measurable function ${\displaystyle X}$ from a probability infinite ${\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}$ to a measurable space ${\displaystyle ({\mathcal {X}},{\mathcal {A}})}$ . Given that probabilities of events of the form ${\displaystyle \{\omega \in \Omega \mid Ten(\omega )\in A\}}$ satisfy Kolmogorov'south probability axioms, the probability distribution of ${\displaystyle X}$ is the pushforward measure ${\displaystyle X_{*}\mathbb {P} }$ of ${\displaystyle 10}$ , which is a probability measure on ${\displaystyle ({\mathcal {X}},{\mathcal {A}})}$ satisfying ${\displaystyle X_{*}\mathbb {P} =\mathbb {P} 10^{-1}}$ .^{[22] [23] [24]}

Other kinds of distributions [edit]

One solution for the Rabinovich–Fabrikant equations. What is the probability of observing a country on a sure place of the support (i.e., the cherry-red subset)?

Admittedly continuous and detached distributions with support on ${\displaystyle \mathbb {R} ^{chiliad}}$ or ${\displaystyle \mathbb {Due north} ^{k}}$ are extremely useful to model a myriad of phenomena,^{[4] [half dozen]} since virtually practical distributions are supported on relatively uncomplicated subsets, such as hypercubes or balls. However, this is not always the instance, and there exist phenomena with supports that are actually complicated curves ${\displaystyle \gamma :[a,b]\rightarrow \mathbb {R} ^{due north}}$ within some space ${\displaystyle \mathbb {R} ^{due north}}$ or similar. In these cases, the probability distribution is supported on the image of such bend, and is likely to exist determined empirically, rather than finding a closed formula for information technology.^[25]

I example is shown in the figure to the correct, which displays the evolution of a system of differential equations (ordinarily known as the Rabinovich–Fabrikant equations) that can be used to model the behaviour of Langmuir waves in plasma.^[26] When this phenomenon is studied, the observed states from the subset are equally indicated in reddish. So one could inquire what is the probability of observing a state in a certain position of the ruby subset; if such a probability exists, it is called the probability measure of the arrangement.^{[27] [25]}

This kind of complicated support appears quite oftentimes in dynamical systems. It is not elementary to constitute that the organization has a probability measure, and the main problem is the following. Let ${\displaystyle t_{ane}\ll t_{two}\ll t_{3}}$ be instants in time and ${\displaystyle O}$ a subset of the back up; if the probability measure exists for the system, one would expect the frequency of observing states inside set ${\displaystyle O}$ would be equal in interval ${\displaystyle [t_{ane},t_{2}]}$ and ${\displaystyle [t_{two},t_{iii}]}$ , which might not happen; for example, information technology could oscillate similar to a sine, ${\displaystyle \sin(t)}$ , whose limit when ${\displaystyle t\rightarrow \infty }$ does non converge. Formally, the measure exists only if the limit of the relative frequency converges when the system is observed into the infinite future.^[28] The branch of dynamical systems that studies the being of a probability measure out is ergodic theory.

Note that even in these cases, the probability distribution, if it exists, might notwithstanding be termed "absolutely continuous" or "detached" depending on whether the support is uncountable or countable, respectively.

Random number generation [edit]

Most algorithms are based on a pseudorandom number generator that produces numbers ${\displaystyle X}$ that are uniformly distributed in the half-open interval [0, ane). These random variates ${\displaystyle X}$ are and so transformed via some algorithm to create a new random variate having the required probability distribution. With this source of compatible pseudo-randomness, realizations of any random variable can be generated.^[29]

For example, suppose ${\displaystyle U}$ has a uniform distribution betwixt 0 and one. To construct a random Bernoulli variable for some ${\displaystyle 0<p<one}$ , we ascertain

$${\displaystyle 10={\begin{cases}1,&{\text{if }}U<p\\0,&{\text{if }}U\geq p\finish{cases}}}$$

so that

${\displaystyle \Pr(X=1)=\Pr(U<p)=p,\quad \Pr(10=0)=\Pr(U\geq p)=1-p.}$

This random variable X has a Bernoulli distribution with parameter ${\displaystyle p}$ .^[29] Notation that this is a transformation of discrete random variable.

For a distribution office ${\displaystyle F}$ of an admittedly continuous random variable, an absolutely continuous random variable must be constructed. ${\displaystyle F^{\mathit {inv}}}$ , an inverse function of ${\displaystyle F}$ , relates to the uniform variable ${\displaystyle U}$ :

$${\displaystyle {U\leq F(x)}={F^{\mathit {inv}}(U)\leq x}.}$$

For example, suppose a random variable that has an exponential distribution ${\displaystyle F(ten)=i-e^{-\lambda x}}$ must be constructed.

$${\displaystyle {\begin{aligned}F(10)=u&\Leftrightarrow 1-eastward^{-\lambda x}=u\\[2pt]&\Leftrightarrow eastward^{-\lambda ten}=1-u\\[2pt]&\Leftrightarrow -\lambda x=\ln(one-u)\\[2pt]&\Leftrightarrow x={\frac {-one}{\lambda }}\ln(i-u)\end{aligned}}}$$

so ${\displaystyle F^{\mathit {inv}}(u)={\frac {-1}{\lambda }}\ln(1-u)}$ and if ${\displaystyle U}$ has a ${\displaystyle U(0,1)}$ distribution, and so the random variable ${\displaystyle X}$ is defined by ${\displaystyle X=F^{\mathit {inv}}(U)={\frac {-one}{\lambda }}\ln(1-U)}$ . This has an exponential distribution of ${\displaystyle \lambda }$ .^[29]

A frequent problem in statistical simulations (the Monte Carlo method) is the generation of pseudo-random numbers that are distributed in a given way.

Common probability distributions and their applications [edit]

The concept of the probability distribution and the random variables which they describe underlies the mathematical discipline of probability theory, and the scientific discipline of statistics. There is spread or variability in almost any value that can exist measured in a population (due east.g. height of people, immovability of a metal, sales growth, traffic flow, etc.); nearly all measurements are made with some intrinsic error; in physics, many processes are described probabilistically, from the kinetic properties of gases to the quantum mechanical clarification of primal particles. For these and many other reasons, unproblematic numbers are often inadequate for describing a quantity, while probability distributions are often more appropriate.

The post-obit is a list of some of the well-nigh common probability distributions, grouped by the type of process that they are related to. For a more complete list, see list of probability distributions, which groups by the nature of the outcome being considered (discrete, admittedly continuous, multivariate, etc.)

All of the univariate distributions below are singly peaked; that is, it is causeless that the values cluster around a single point. In practice, actually observed quantities may cluster effectually multiple values. Such quantities tin can be modeled using a mixture distribution.

Linear growth (e.g. errors, offsets) [edit]

• Normal distribution (Gaussian distribution), for a single such quantity; the most commonly used admittedly continuous distribution

Exponential growth (e.one thousand. prices, incomes, populations) [edit]

• Log-normal distribution, for a single such quantity whose log is unremarkably distributed
• Pareto distribution, for a single such quantity whose log is exponentially distributed; the prototypical ability law distribution

Uniformly distributed quantities [edit]

• Discrete compatible distribution, for a finite set of values (e.yard. the upshot of a fair die)
• Continuous uniform distribution, for absolutely continuously distributed values

Bernoulli trials (aye/no events, with a given probability) [edit]

• Basic distributions:
  • Bernoulli distribution, for the outcome of a single Bernoulli trial (e.g. success/failure, yeah/no)
  • Binomial distribution, for the number of "positive occurrences" (e.one thousand. successes, yeah votes, etc.) given a fixed total number of independent occurrences
  • Negative binomial distribution, for binomial-type observations merely where the quantity of involvement is the number of failures earlier a given number of successes occurs
  • Geometric distribution, for binomial-type observations but where the quantity of interest is the number of failures before the starting time success; a special case of the negative binomial distribution
• Related to sampling schemes over a finite population:
  • Hypergeometric distribution, for the number of "positive occurrences" (e.thousand. successes, yeah votes, etc.) given a fixed number of full occurrences, using sampling without replacement
  • Beta-binomial distribution, for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed number of full occurrences, sampling using a Pólya urn model (in some sense, the "opposite" of sampling without replacement)

Categorical outcomes (events with 1000 possible outcomes) [edit]

• Categorical distribution, for a single categorical issue (east.g. yeah/no/perhaps in a survey); a generalization of the Bernoulli distribution
• Multinomial distribution, for the number of each type of categorical effect, given a fixed number of total outcomes; a generalization of the binomial distribution
• Multivariate hypergeometric distribution, similar to the multinomial distribution, but using sampling without replacement; a generalization of the hypergeometric distribution

Poisson process (events that occur independently with a given rate) [edit]

• Poisson distribution, for the number of occurrences of a Poisson-blazon event in a given flow of fourth dimension
• Exponential distribution, for the time before the next Poisson-type outcome occurs
• Gamma distribution, for the time before the next 1000 Poisson-type events occur

Absolute values of vectors with usually distributed components [edit]

• Rayleigh distribution, for the distribution of vector magnitudes with Gaussian distributed orthogonal components. Rayleigh distributions are found in RF signals with Gaussian real and imaginary components.
• Rice distribution, a generalization of the Rayleigh distributions for where there is a stationary background point component. Found in Rician fading of radio signals due to multipath propagation and in MR images with racket corruption on non-zero NMR signals.

Normally distributed quantities operated with sum of squares [edit]

• Chi-squared distribution, the distribution of a sum of squared standard normal variables; useful e.g. for inference regarding the sample variance of normally distributed samples (meet chi-squared exam)
• Student's t distribution, the distribution of the ratio of a standard normal variable and the square root of a scaled chi squared variable; useful for inference regarding the mean of normally distributed samples with unknown variance (see Student'due south t-examination)
• F-distribution, the distribution of the ratio of ii scaled chi squared variables; useful e.g. for inferences that involve comparing variances or involving R-squared (the squared correlation coefficient)

As cohabit prior distributions in Bayesian inference [edit]

• Beta distribution, for a single probability (real number between 0 and 1); conjugate to the Bernoulli distribution and binomial distribution
• Gamma distribution, for a non-negative scaling parameter; conjugate to the rate parameter of a Poisson distribution or exponential distribution, the precision (inverse variance) of a normal distribution, etc.
• Dirichlet distribution, for a vector of probabilities that must sum to i; cohabit to the chiselled distribution and multinomial distribution; generalization of the beta distribution
• Wishart distribution, for a symmetric non-negative definite matrix; conjugate to the inverse of the covariance matrix of a multivariate normal distribution; generalization of the gamma distribution^[30]

Some specialized applications of probability distributions [edit]

• The cache language models and other statistical linguistic communication models used in tongue processing to assign probabilities to the occurrence of item words and word sequences do so by means of probability distributions.
• In quantum mechanics, the probability density of finding the particle at a given indicate is proportional to the square of the magnitude of the particle's wavefunction at that point (come across Born rule). Therefore, the probability distribution function of the position of a particle is described past ${\textstyle P_{a\leq x\leq b}(t)=\int _{a}^{b}dx\,|\Psi (x,t)|^{2}}$ , probability that the particle's position 10 will be in the interval a ≤ x ≤ b in dimension one, and a similar triple integral in dimension three. This is a key principle of breakthrough mechanics.^[31]
• Probabilistic load menses in power-period report explains the uncertainties of input variables as probability distribution and provides the power flow adding too in term of probability distribution.^[32]
• Prediction of natural phenomena occurrences based on previous frequency distributions such as tropical cyclones, hail, fourth dimension in between events, etc.^[33]

See as well [edit]

• Conditional probability distribution
• Joint probability distribution
• Quasiprobability distribution
• Empirical probability distribution
• Histogram
• Riemann–Stieltjes integral application to probability theory

Lists [edit]

• List of probability distributions
• Listing of statistical topics

References [edit]

Citations [edit]

1. ^ ^{a b} Everitt, Brian (2006). The Cambridge dictionary of statistics (3rd ed.). Cambridge, UK: Cambridge Academy Press. ISBN978-0-511-24688-3. OCLC 161828328.
2. ^ Ash, Robert B. (2008). Basic probability theory (Dover ed.). Mineola, Due north.Y.: Dover Publications. pp. 66–69. ISBN978-0-486-46628-6. OCLC 190785258.
3. ^ ^{a b} Evans, Michael; Rosenthal, Jeffrey S. (2010). Probability and statistics: the scientific discipline of uncertainty (second ed.). New York: West.H. Freeman and Co. p. 38. ISBN978-1-4292-2462-8. OCLC 473463742.
4. ^ ^{a b c d eastward} Ross, Sheldon M. (2010). A first course in probability. Pearson.
5. ^ ^{a b} "1.3.6.1. What is a Probability Distribution". www.itl.nist.gov . Retrieved 2020-09-10 .
6. ^ ^{a b} A modernistic introduction to probability and statistics : understanding why and how. Dekking, Michel, 1946-. London: Springer. 2005. ISBN978-1-85233-896-1. OCLC 262680588. {{cite volume}}: CS1 maint: others (link)
7. ^ ^{a b} Chapters ane and 2 of Vapnik (1998)
8. ^ Walpole, R.E.; Myers, R.H.; Myers, S.L.; Ye, Yard. (1999). Probability and statistics for engineers. Prentice Hall.
9. ^ ^{a b} DeGroot, Morris H.; Schervish, Mark J. (2002). Probability and Statistics. Addison-Wesley.
10. ^ Billingsley, P. (1986). Probability and measure out. Wiley. ISBN9780471804789.
11. ^ Shephard, N.K. (1991). "From characteristic part to distribution function: a uncomplicated framework for the theory". Econometric Theory. 7 (4): 519–529. doi:ten.1017/S0266466600004746.
12. ^ ^{a b} More data and examples can be plant in the articles Heavy-tailed distribution, Long-tailed distribution, fatty-tailed distribution
13. ^ Erhan, Çınlar (2011). Probability and stochastics. New York: Springer. p. 57. ISBN9780387878584.
14. ^ see Lebesgue's decomposition theorem
15. ^ Erhan, Çınlar (2011). Probability and stochastics. New York: Springer. p. 51. ISBN9780387878591. OCLC 710149819.
16. ^ Cohn, Donald Fifty. (1993). Measure theory. Birkhäuser.
17. ^ Khuri, André I. (March 2004). "Applications of Dirac's delta function in statistics". International Journal of Mathematical Instruction in Scientific discipline and Engineering. 35 (2): 185–195. doi:10.1080/00207390310001638313. ISSN 0020-739X. S2CID 122501973.
18. ^ Fisz, Marek (1963). Probability Theory and Mathematical Statistics (tertiary ed.). John Wiley & Sons. p. 129. ISBN0-471-26250-1.
19. ^ Jeffrey Seth Rosenthal (2000). A First Look at Rigorous Probability Theory. World Scientific.
20. ^ Affiliate three.two of DeGroot & Schervish (2002)
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• den Dekker, A. J.; Sijbers, J. (2014). "Information distributions in magnetic resonance images: A review". Physica Medica. thirty (7): 725–741. doi:10.1016/j.ejmp.2014.05.002. PMID 25059432.
• Vapnik, Vladimir Naumovich (1998). Statistical Learning Theory. John Wiley and Sons.

External links [edit]

• "Probability distribution", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Field Guide to Continuous Probability Distributions, Gavin E. Crooks.

What Is The Term Used To Describe The Distribution Of A Data Set With One Mode,

Source: https://en.wikipedia.org/wiki/Probability_distribution

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